Let's break down the problem step by step:

Step 1: Define variables

Let:

• $x$ be the number of hours it takes the second hose to fill the pool by itself.
• The first hose can fill the pool in 4 hours.

Step 2: Rate of filling the pool

The rate at which each hose fills the pool is the reciprocal of the time it takes to fill the pool:

• The rate of the first hose is $\frac{1}{4}$ of the pool per hour.
• The rate of the second hose is $\frac{1}{x}$ of the pool per hour.

Step 3: Combined rate of both hoses

When both hoses are working together, their combined rate is the sum of their individual rates: $\frac{1}{4} + \frac{1}{x}$

Step 4: Rate of the second hose compared to the two hoses combined

We are told that the second hose takes 2 more hours than the combined rate of both hoses. This means that: $x = \left( 4 \, \text{hours} + 2 \right) \quad \text{(time for the second hose to fill the pool by itself)}$ So, we have the equation: $x = 6 \, \text{hours}$

Step 5: Conclusion

It would take the second hose 6 hours to fill the pool by itself.

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Would you like further details on the solution? Or is there anything specific you'd like clarified?

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Additional Questions for Learning:

1. How do we define and work with rates in related rate problems?
2. How do you approach problems involving combined rates or work rates?
3. What is the significance of the reciprocal when dealing with rates of work or filling problems?
4. Can this method be applied to other types of work problems, such as multiple workers or machines?
5. What happens if we change the number of hours the first hose takes to fill the pool?

Tip: Always check that the units for time and rate match up properly before proceeding with the solution to avoid confusion!